To unify the discussion of (REF )-(), we adopt the
language of the finite element exterior calculus (FEEC, [1]}). We
recognize functions in \(H(\operatorname{grad})\) , \(H(\operatorname{curl})\) , \(H(\operatorname{div})\) , and \(L^2\)  as differential
\(k\) -forms for \(k = 0, 1, 2, 3\)  respectively, writing \(H(\operatorname{d}^k, \Omega ) = H\Lambda ^k(\Omega )\) , with \(\operatorname{d}^0 = \operatorname{grad}\) , \(\operatorname{d}^1 = \operatorname{curl}\) , \(\operatorname{d}^2 = \operatorname{div}\) , \(\operatorname{d}^3 = \operatorname{null}\)  (the zero map). We
define the spaces
\(V^k \lbrace  v \in H \Lambda ^k(\Omega ) : \operatorname{tr}v = 0 \mbox{~on~} \Gamma _D\rbrace ,\) 
